Difference between revisions of "Math 360, Fall 2021, Assignment 11"
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Revision as of 20:36, 21 November 2021
The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
- - Saint Augustine
Read:
- Section 9.
Carefully define the following terms, then give one example and one non-example of each:
- Fixed point (of a permutation $\pi$).
- Moving point (of a permutation $\pi$).
- Disjoint (permutations $\pi$ and $\sigma$).
- Orbit (of a permutation $\pi$).
- Cycle.
- $(i_1,\dots,i_k)$ (the cycle determined by the sequence $i_1,\dots,i_k$).
- Length (of a cycle).
Carefully state the following theorems (you do not need to prove them):
- Theorem relating $\sigma\tau$ to $\tau\sigma$, when $\sigma$ and $\tau$ are disjoint.
- Theorem concerning disjoint cycle decomposition.
Carefully practice the following calculations, giving a worked example of each:
- Conversion of two-row notation to cycle notation.
- Conversion of cycle notation to two-row notation.
- Composition of two permutations, expressed in cycle notation.
- Inversion of a permutation, expressed in cycle notation.
Solve the following problems:
- Section 9, problems 1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16. (Hint for 14-16: recall that the order of a finite group element $g$ is the least positive integer $n$ with $g^n=1$. For a permutation, this will be the least common multiple of the lengths of the cycles in its disjoint cycle decomposition.)